We propose a robust and eicient field-aligned volumetric meshing algorithm that produces hex-dominant meshes, i.e. meshes that are predominantly composed of hexahedral elements while containing a small number of irregular polyhedra. The latter are placed according to the singularities of two optimized guiding fields, which allow our method to generate meshes with an exceptionally high amount of isotropy. The field design phase of our method relies on a compact quaternionic representation of volumetric octa-ields and a corresponding optimization that explicitly models the discrete matchings between neighboring elements. This optimization naturally supports alignment constraints and scales to very large datasets. We also propose a novel extraction technique that uses field-guided mesh simpliication to convert the optimized ields into a hexdominant output mesh. Each simpliication operation maintains topological validity as an invariant, ensuring manifold output. These steps easily generalize to other dimensions or representations, and we show how they can be an asset in existing 2D surface meshing techniques. Our method can automatically and robustly convert any tetrahedral mesh into an isotropic hex-dominant mesh and (with minor modiications) can also convert any triangle mesh into a corresponding isotropic quad-dominant mesh, preserving its genus, number of holes, and manifoldness. We demonstrate the beneits of our algorithm on a large collection of shapes provided in the supplemental material along with all generated results.